C.1.1 General
For instruments classified for interpolation, the calibration uncertainty is the uncertainty in the force value calculated from the interpolation equation, at any deflection, for increasing forces only. For instruments classified for specific forces only, the calibration uncertainty is the uncertainty in the force corresponding to any deflection equal to one of the mean deflections obtained during the calibration, for increasing forces only.
At each calibration force, F, a combined standard uncertainty, uc, expressed in units of force, is calculated from the readings obtained during the calibration. These combined standard uncertainties are plotted against force, and a least-squares fit covering all the values is calculated. The coefficients of this fit are then multiplied by the coverage factor k = 2 to give an expanded uncertainty value, U, for any force within the calibration range. An expanded relative uncertainty value, W, can also then be calculated.
8 2 c
1 i i
u u
=
= ∑ (C.1)
and
U = k × uc (C.2)
and
W = U/F (C.3)
where
u1 is the standard uncertainty associated with the applied calibration force;
u2 is the standard uncertainty associated with the reproducibility of the calibration results;
u3 is the standard uncertainty associated with the repeatability of the calibration results;
u4 is the standard uncertainty associated with the resolution of the indicator;
u5 is the standard uncertainty associated with the creep of the instrument;
u6 is the standard uncertainty associated with the drift in zero output;
u7 is the standard uncertainty associated with temperature of instrument;
u8 is the standard uncertainty associated with interpolation.
The expanded relative uncertainty, W, may also be calculated from a combination of relative standard uncertainties, wi:
8 2
c 1
i i
w w
=
= ∑ (C.4)
and
W = k × wc (C.5)
and
U = W × F (C.6)
where
w1 is the relative standard uncertainty associated with applied calibration force;
w2 is the relative standard uncertainty associated with reproducibility of calibration results;
w3 is the relative standard uncertainty associated with repeatability of calibration results;
w4 is the relative standard uncertainty associated with resolution of indicator;
w5 is the relative standard uncertainty associated with creep of instrument;
w6 is the relative standard uncertainty associated with the drift in the zero output;
w7 is the relative standard uncertainty associated with the temperature of the instrument;
w8 is the relative standard uncertainty associated with interpolation.
NOTE 1 The interpolation component (u8, w8) is not taken into account in the calibration uncertainty with instruments classified for specific forces only.
NOTE 2 The relative uncertainty can be expressed as a percentage by multiplying by 100.
C.1.2 Calculation of calibration force uncertainty, u1, w1
u1 is simply the standard uncertainty associated with the forces applied by the calibration machine, expressed in units of force, and w1 is the same expressed as a relative value.
C.1.3 Calculation of reproducibility uncertainty, u2, w2
u2 is the standard deviation associated with the mean incremental deflections obtained during the calibration, expressed in units of force, and w2 is the same expressed as a relative value.
( )2
2 N r
N 1,3, 5
1
6 i i
u F X X
X =
= × × ∑ − (C.7)
and
( )2
2 r
r 1,3, 5
1 1
6 i i
w X X
X =
= × × ∑ − (C.8)
where Xi is the deflection obtained in the incremental series 1, 3, and 5.
NOTE This is not the reproducibility of a measured force during the force-proving instrument's subsequent use.
C.1.4 Calculation of repeatability uncertainty, u3, w3
u3 is the uncertainty contribution due to the repeatability of the measured deflection, expressed in units of force, and w3 is the same expressed as a relative value. It can be assumed that, at each force, F:
3 100 3
b F
u ′×
= × (C.9)
and
3 100 3
w = b′
× (C.10)
where b′ is the relative repeatability error as defined in 7.5.1.
C.1.5 Calculation of resolution uncertainty, u4, w4
Each deflection value is calculated from two readings (the reading with an applied force minus the reading at zero force). Because of this, the resolution of the indicator needs to be included twice as two rectangular distributions, each with a standard uncertainty of r (2 3 ), where r is the resolution, expressed in units of force:
4 6
u = r (C.11)
and
4
1 6 w r
= ×F (C.12)
C.1.6 Calculation of creep uncertainty, u5, w5
This uncertainty component is due to the fact that, at a given force, the measured deflection can be influenced by the previous short-term loading history. One measure of this influence is the change in transducer output in the period from 30 s to 300 s after application or removal of the maximum calibration force. This effect is not included in the reproducibility because, generally, the same calibration machine is used for all measurement series and the time/loading profile will be the same.
This effect can be estimated as follows:
5 100 3
c F u = ×
× (C.13)
and
5 100 3
w = c
× (C.14)
If the creep test is not performed, the creep uncertainty can be estimated by dividing the hysteresis by a factor of 3. Therefore, the following equation can be used to calculate this uncertainty contribution for increasing forces:
5 100 3 3 u = ν×F
× (C.15)
and
5 100 3 3
w = ν
× (C.16)
C.1.7 Calculation of zero drift uncertainty, u6, w6
This uncertainty component is due to the fact that the instrument's zero output can vary between measurement series and that the measured deflections could be a function of the time spent at zero force between series. This effect is not included in reproducibility because, generally, this time will be the same for all measurement series. One measure of this variation is the zero error, f0, so this effect can be estimated as follows:
0
6 100
f F
u = × (C.17)
and
6 0
100
w = f (C.18)
C.1.8 Calculation of temperature uncertainty, u7, w7
This uncertainty is the contribution due to the variation of temperature throughout the calibration, together with the uncertainty in the measurement of the calibration temperature. The sensitivity of the instrument to temperature needs to be estimated, either by tests or from the manufacturer's specifications or by theory or by experience. Expressing this component in units of force or as a relative value, we have:
7
1
2 3
u = ×K ∆T × ×F (C.19)
and
7
1
2 3
w = ×K ∆T × (C.20)
where
K is the instrument's temperature coefficient, in reciprocal degrees Celsius (°C−1);
∆T is the calibration temperature range, allowing for the uncertainty in the measurement of the temperature.
C.1.9 Calculation of interpolation uncertainty, u8, w8 C.1.9.1 General
This component is not taken into account in the calibration uncertainty for instruments classified for specific forces only.
It is the contribution due to the plotted force/deflection points not all falling on the best-fit line, leading to an uncertainty in the interpolation equation. Either of the two methods given in C.1.9.2 and C.1.9.3 may be used to calculate this contribution.
C.1.9.2 Residual method
The component can be estimated using statistical theory. Assuming that the calibration forces are evenly distributed, it can be simplified to the following equations:
N r
8
N 1
u F
X n d
= δ
− − (C.21)
and
N r
8
N 1
w F
F X n d
= δ
× − − (C.22)
where
δr is the sum of the squared deviations between the mean deflection and the value calculated from the interpolation equation;
n is the number of force calibration steps;
d is the degree of the interpolation equation.
C.1.9.3 Deviation method
The component is the difference between the mean measured deflection and the value calculated from the interpolation equation:
r a
8
r
X X
u F
X
= − × (C.23)
or
r a
8
r
X X
w X
= − (C.24)
C.1.10 Calculation of combined calibration standard uncertainty and of expanded uncertainty
C.1.10.1 General
The calculation of the combined standard uncertainty and expanded uncertainty is carried out either in force units (for uc and U) or as relative values (for wc and W), as shown in C.1.10.2 and C.1.10.3, respectively.
C.1.10.2 Combined standard uncertainty and expanded uncertainty in force units
For each calibration force, calculate the combined standard uncertainty, uc, by combining the individual standard uncertainties in quadrature:
8 2 c
1 i i
u u
=
= ∑ (C.25)
NOTE This equation is the same as Equation (C.1).
Plot a graph of uc against force and then determine the coefficients of the best-fit least-squares line through all of the data points.
The form of the fitted line (e.g. linear, polynomial, exponential) will depend on the calibration results. A linear equation is preferred for reasons of simplicity. If this results in values lower than the minimum combined uncertainty value, a more conservative fit should be employed and/or a minimum value for the uncertainty should be specified. It is suggested that this be equal to the minimum combined standard uncertainty obtained (in units of force).
The expanded uncertainty, U, is given by the equation whose coefficients are twice those of the best-fit equation. For any force within the calibration range, an expanded uncertainty can then be calculated, expressed in force units.
C.1.10.3 Combined standard uncertainty and expanded uncertainty as relative values
For each calibration force, calculate the combined standard uncertainty, wc, by combining the individual standard uncertainties in quadrature:
8 c 2
1 i i
w w
=
= ∑ (C.26)
NOTE This equation is the same as Equation (C.4).
Plot a graph of wc against force and then determine the coefficients of the best-fit least-squares line through all of the data points.
The form of the fitted line (e.g. linear, polynomial, exponential) will depend on the calibration results. If this results in values lower than the minimum combined uncertainty value, a more conservative fit should be employed and/or a minimum value for the uncertainty should be specified for the relevant parts of the calibration range. It is suggested that this be equal to the minimum combined standard uncertainty obtained.
The expanded uncertainty, W, is given by the equation whose coefficients are twice those of the best-fit equation. For any force within the calibration range, an expanded uncertainty can then be calculated, expressed as a relative value.